Criteria for the Stochastic Ordering of Random Sums,with Actuarial Applications

It is shown that vectors ( S M 1 , … , S Mn ) and ( S' M'1 , …, S' M'n ) of random sums of positive random variables are stochastically ordered by upper orthant dependence, lower orthant dependence, concordance or by the supermodular ordering whenever their corresponding random numbers of terms ( M 1 , … , M n ) and ( M' 1 , … , M' n ) are themselves ordered in this fashion. Actuarial applications of these results are given to different dependence structures for the collective risk model with several classes of business.     

The multivariate Variance Gamma model: basket option pricing and calibration

In this paper, we propose a methodology for pricing basket options in the multivariate Variance Gamma model introduced in Luciano and Schoutens [Quant. Finance 6(5), 385–402]. The stock prices composing the basket are modelled by time-changed geometric Brownian motions with a common Gamma subordinator. Using the additivity property of comonotonic stop-loss premiums together with Gauss-Laguerre polynomials, we express the basket option price as a linear combination of Black & Scholes prices. Furthermore, our new basket option pricing formula enables us to calibrate the multivariate VG model in a fast way. As an illustration, we show that even in the constrained situation where the pairwise correlations between the Brownian motions are assumed to be equal, the multivariate VG model can closely match the observed Dow Jones index options.     

Some problems in actuarial finance involving sums of dependent risks

A trend in actuarial finance is to combine technical risk with interest risk. If Y_t , t = 1, 2, denotes the timevalue of money (discount factors at time t ) and X_t the stochastic payments to be made at time t , the random variable of interest is often the scalar product of these two random vectors V = X_t Y_t . The vectors X and Y are supposed to be independent, although in general they have dependent components. The current insurance practice based on the law of large numbers disregards the stochastic financial aspects of insurance. On the other hand, introduction of the variables Y ₁, Y ₂, to describe the financial aspects necessitates estimation or knowledge of their distribution function.
We investigate some statistical models for problems of insurance and finance, including Risk Based Capital/Value at Risk, Asset Liability Management, the distribution of annuities, cash flow evaluations (in the framework of pension funds, embedded value of a portfolio, Asian options) and provisions for claims incurred, but not reported (IBNR).     

Stochastic modelling of herd behaviour indices

This paper proposes different diffusion processes to model herd behaviour indices such as the Herd Behaviour Index (HIX). These models arise by combining popular mean-reverting processes with simple algebraic functions mapping the definition domain of the underlying mean-reverting process to the unit interval. The so obtained Itô processes preserve, to some extent, the mean-reverting trend of the underlying process while satisfying the fundamental properties of the so-called herd behaviour indices. In a numerical study, we calibrate the different model settings to time series data for a period spanning from January 2000 until October 2009 and investigate their ability to predict the future behaviour of herd behaviour indices.     

On the distribution of discounted loss reserves using generalized linear models

Renshaw and Verrall [] specified the generalized linear model (GLM) underlying the chain-ladder technique and suggested some other GLMs which might be useful in claims reserving. The purpose of this paper is to construct bounds for the discounted loss reserve within the framework of GLMs. Exact calculation of the distribution of the total reserve is not feasible, and hence the determination of lower and upper bounds with a simpler structure is a possible way out. The paper ends with numerical examples illustrating the usefulness of the presented approximations.     

On Tail Value-at-Risk for sums of non-independent random variables with a generalized Pareto distribution

Recently in actuarial literature several authors have derived lower and upper bounds in the sense of convex order for sums of random variables with given marginal distributions and unknown dependency structure. In this paper, we derive convex bounds for sums of non-independent and identically distributed random variables when marginal distributions are mixture models. In particular, we examine some well-known risk measures and we find approximations for Tail Value-at-Risk of the sums considered when marginal distributions are generalized Pareto distributions. By numerical examples we illustrate the goodness of the presented approximations.      

American-type basket option pricing: a simple two-dimensional partial differential equation

We consider the pricing of American-type basket derivatives by numerically solving a partial differential equation (PDE). The curse of dimensionality inherent in basket derivative pricing is circumvented by using the theory of comonotonicity. We start with deriving a PDE for the European-type comonotonic basket derivative price, together with a unique self-financing hedging strategy. We show how to use the results for the comonotonic market to approximate American-type basket derivative prices for a basket with correlated stocks. Our methodology generates American basket option prices which are in line with the prices obtained via the standard Least-Square Monte-Carlo approach. Moreover, the numerical tests illustrate the performance of the proposed method in terms of computation time, and highlight some deficiencies of the standard LSM method.